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Integration (p-1) — Maths (commerce) STD 12 Commerce / Arts — Question

Maharashtra BoardEnglish MediumSTD 12 Commerce / ArtsMaths (commerce)Integration (p-1)2 Marks
Question
Evalute : $\int x e^{2 x} d x$
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Answer

$
\begin{aligned}
\int x e^{2 x} d x & =x \int e^{2 x} d x-\int\left[\frac{d}{d x}(x) \int e^{2 x} d x\right] d x \\
& =x \cdot \frac{e^{2 x}}{2}-\int 1 \cdot \frac{e^{2 x}}{2} d x \\
& =\frac{1}{2} x e^{2 x}=\frac{1}{2} \int e^{2 x} d x \\
& =\frac{1}{2} x e^{2 x}-\frac{1}{2} \cdot \frac{e^{2 x}}{2}+c \\
& =e^{2 x}\left(\frac{x}{2}-\frac{1}{4}\right)+c \\
& =\left(\frac{2 x-1}{4}\right) e^{2 x}+c .
\end{aligned}
$

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